#69726

order	384
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1452

Subgroups

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₃
commutator, socle	C₂²×C₃	C₂×C₄²
Fitting	C₂×C₃×C₄×C₈	C₂
prime radical	C₂³×C₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#1452
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3)(14,20)(17,23)(21,26)(24,27);(2,3)(4,5,6,9,7,8,10,11)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(5,8)(9,11)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)

Polycyclic presentation:

2^7,3|a^2=d,b^2=e,d^2=f,e^2=g,b^a=bg,c^b=cf,h^b=h^2

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682039, #10682041, #10682165, …
maximal normal subgroup of	#10682039, #10682041, #10682165, …
maximal quotient of	#10684552, #10684572, #10685616, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 2302);